When your predictor variables (in this case anxiety and depression) are highly correlated (which r=0.8 might suggest), then you approach the multicolinearity situation, i.e. where your variables are so highly correlated that they can be predicted from each other with a reasonable degree of accuracy.

Multiple regression models like the one you are suggesting run into difficulties when your predictors are multicolinear such as loss of power and uninterpretable beta coefficients (not to mention that it calls into question the experimental relevance of including it in the model in the first place) and thus absence of multicolinearity is one of the assumptions.

If you believe this is the case for your variables (which it seems it may be, although you can ask SPSS for multicolinearity estimates), you would not want to run a standard multiple regression model. Your options are:

1) Simply remove one of your variables from the model

2) Somehow construct a weighted composite score of anxiety and depression if this is relevant in the real world

3) Construct a partial least squares regression model instead, which is able to deal with multicolinearity and can be run in SPSS (or R if you're technically minded).

The third option is probably the best way to get the most out of your data, if you think a composite score makes sense in a real world way then great and just run a standard binomial linear regression, or if actually you think one of your variables is unimportant to your model then omit it, but I don't think this is such a good idea.

I hope that helps/makes sense!

When your predictor variables (in this case anxiety and depression) are highly correlated (which r=0.8 might suggest), then you approach the multicolinearity situation, i.e. where your variables are so highly correlated that they can be predicted from each other with a reasonable degree of accuracy.

Multiple regression models like the one you are suggesting run into difficulties when your predictors are multicolinear such as loss of power and uninterpretable beta coefficients (not to mention that it calls into question the experimental relevance of including it in the model in the first place) and thus absence of multicolinearity is one of the assumptions.

If you believe this is the case for your variables (which it seems it may be, although you can ask SPSS for multicolinearity estimates), you would not want to run a standard multiple regression model. Your options are:

1) Simply remove one of your variables from the model

2) Somehow construct a weighted composite score of anxiety and depression if this is relevant in the real world

3) Construct a partial least squares regression model instead, which is able to deal with multicolinearity and can be run in SPSS (or R if you're technically minded).

The third option is probably the best way to get the most out of your data, if you think a composite score makes sense in a real world way then great and just run a standard binomial linear regression, or if actually you think one of your variables is unimportant to your model then omit it, but I don't think this is such a good idea.

I hope that helps/makes sense!

In the absence of an a-priori hypothesis involving either anxiety or depression, an aggregate score (as suggested by the second suggestion in the comment above) might suffice. Yet, you might consider the limitations of your measures and methodology. Were the measures self-reports or clinical interviews?

As anxiety and depression are strongly correlated they will not be independent predictors, you can rather talk about them as the "anxiety-depression pair". When you include both in your model other predictors will be adjusted for the "pair". However, you will not be able to assess the independent effect of anxiety only or depression only.

　To learn more about the effect of anxiety or depression you could substitute them with their two principal components. The first (strong) PC will contain the common aspect of anxiety and depression. The second (weak) PC will measure the dissimilarity.

If they present a correlation of 0.8, probably they are not independent and, as stated above, you will face a multicolinearity problem. You can try the "weighted composite" proposed by Hefin Jones using principal component or factor analysis.

Could you elaborate on the response variable. How are you measuring quality of sleep? How are you measuring anxiety and depression? How did you establish an association? I need to know more about the experiment before making any recommendations. What was the sample size?

You should definitely check the following very recent paper dealing with the situation of colinearity of predictors and what to do in these cases:

Wurm, L. H., & Fisicaro, S. A. (2014). What residualizing predictors in regression analyses does (and what it does not do). Journal of Memory and Language, 72, 37–48. doi:10.1016/j.jml.2013.12.003

link: http://clasweb.clas.wayne.edu/Multimedia/wurm/files/Wurm%20%26%20Fisicaro%20(2014).pdf

What it will tell you is that if you enter them simultaneously, neither of them will include the shared variance between them but the significance test for the parameters will only tell you if there is unique variance to them. So if you would like to see the influence of the shared (i.e., common) variance among the two predictors you need to enter one before the other (the second model with both then corresponds to the one when you directly enter both).

I think it is necessary start for more simple view, so I am going just pointing out some idea in this way.

Regards what you mention about your studies, both anxiety and depression look they are a response to quality of the sleep. So, the correlation you talked between these two factor just is telling you they response in the same pattern to quality of the sleep. The main first question that is coming my mind is: is that true? both are a consequence of quality of sleep or depression is a consequence of anxiety or vice versa. To me, that is going two define if you should correlate these factor nor shouldn't.

I believe that start for doing a covariance analysis between quality of the sleep, anxiety and depression may give you a simple answer this first question, and it is going to enable you resolving your own question.

I largely agree with Hefin's answer, though whether collinearity really is an issue or not also depends on what the goals of your analysis are. If you only want to predict the DV (sleep quality), then collinearity of the predictors is not a big problem (the more predictors you have - correlated or not - the better the prediction).

However, since you are interested in each predictor's independent contribution, and maybe even want to find out which one of the two (anxiety or depression) is the 'better' predictor, collinearity is indeed a problem. I would probably run separate models (one including anxiety and one including depression) and compare them in terms of fit, i.e. how much variance is explained by each model. Another strategy (although quite fiddly) might be to use a stepwise approach to test different orders in which the two predictors are entered into the model (e.g., if depression still explains some variance after accounting for anxiety, but not the other way round, then depression is probably the better predictor). Or indeed construct a partial least squares model, as Hefin suggested.

Since your variables are highly correlated, there will be multicolinearity if included in the same model.

You may asess the degree of multicolinearity using the VIF (Variance Infalation Factor). The VIF may be calculated for each predictor by doing a linear regression of that predictor on all the other predictors, and then obtaining the R2 from that regression. The VIF is just 1/(1-R2). A VIF of 1.2 tells us that the variance (the square of the standard error) of a particular coefficient is 20% larger than it would be if that predictor was completely uncorrelated with all the other predictors.As a rule of thumb, a variable whose VIF values is greater than 10 may merit further investigation.In SAS you can specify the vif option in Proc Reg to estimate multicolinearity.

As a solution to your problem,if you are measuring anxiety and depression using some of the standard scales such as the Beck Depression Scale or CES-D etc. it may be worthwhile to run a principal components analysis using the original questions comprising the scale to identify the main factors which are common to both variables. The components identified can be used in the regression model instead of the original variables.

Another solution maybe to use centering if you are using scores from a scale to asess depression and anxiety.Transform the two variables by subtracting the mean from each case. The resulting centered data should show lower multicollinearity. A zero b coefficient will now correspond to the independent being at its mean, not at zero, and interpretations of b and beta must be changed accordingly.

From the clinical aspect, anxiety and depression are two different symptoms. The impact and association of these two clinical symptoms with sleep problems is usually different. For example, while depression is associated with early morning insomnia, anxiety is associated with sleep onset insomnia. Therefore, the clinical significance encourages us to consider them as two independent variables simultaneously. However, collinearity needs to be examined as well.

Hi. I would try separate simple linear regressions, one where anxiety predicts sleep quality and one where depression predicts sleep quality. Then I would do the multiple regression including both depression and anxiety as predictors and compare the results (look at the adjusted R-squares). In SAS, you can use the variance inflation factor option in the model statement to check for the influence of collinearity on the two regressors. In SAS, the programming would look like:

Proc reg;

Model sleep_quality = depression sleep/vif;

Look for vif lower than 10 (Kutner MH et al., 2004, Applied Linear Regression Models, 4th edition, McGraw-Hill/Irwin, Boston). Also see the attachment of our recent paper (Lyness SA et al., 10-minute delayed recall from the modified mini-mental state test predicts Alzheimer's disease pathology, Journal of Alzheimer's Disease, [Pre-Press]) where we had this similar challenge (see the "Logistic regression" section in the Results section). Please feel free to email me (and then we can talk on the phone) if you'd like more help.

http://iospress.metapress.com/content/105656/?Content+Status=Accepted&sortorder=asc&v=expanded&o=70

Excellent suggestion given by Hefin. With respect to your study, removing one of the variables from the model or using a weighted composite score would be appropriate as opposed to putting both in the model (suggested in your question). This is because both variables are being evaluated in a sample of chronic medical condition patients and not in depression or anxiety disorder patients. In psychiatric nosology, there can be heterogenous patients with respect to their symptomatology profiles. For example, in depressive disorders, there can be patient with predominantly depressive symptoms as well as depressive patient with comorbid anxiety. Same is true for anxiety disorders. In EEG findings too, topographical differences in activation patterns also differentiate such symptoms/traits. In such scenario, due to heterogeneity found in psychiatric diagnoses, colinearity between anxiety and depression might not be as high as you have found in your study. Your sample group seems more homogeneous and using Hefin's suggestion, a reasonable degree of accuracy in prediction may be expected. A big plus point, you won't lose power and interpretation would be simpler.

This is a great question at so many levels! From a philosophical point of view, the very correlation between these two sets of measurements illustrates, in my mind, that these two entities are not, in the real world, separate things, even though from a DSM point of view, we have been conditioned to see them as separate entities. And from an amateur statistician point of view, teh collinearity between these two measures would be problematic if entered separately. Maybe that is the crux of your question, which a real statistician can answer properly!

Another possibility is to add an interaction term between Anxiety and Depression; possibly using stepwise regression or partial least squares to judge the most influential factors. Correlation is of course not a proof of causation - the interpretation must emanate from basic psychology, not statistics.

In line with your findings, the correlation between anxiety and depression scores is typically high. However, anxiety and depression might be differently associated with different kinds of sleep problems. Typically clinical depression is characterized with early morning wakening whereas uneasiness and anxiety might more typically be associated with problems faling asleep in the evening. If you in your analyses are using self report anxiety/depression measures as continous variables and not as dichotomous variables related to defined cut-off limits as indicators of a clinical anxiety or depressive disorder or not, this differentiation may become less visible. It might be of interest to chech the correlation between anxiety and depression when using recommended cut-off values and then analyse the degree to which these variables are predicting different kinds of sleep problems..

I first want to know whether your variables are categorical or continuous? If they are categorical then what type of correlation are you talking about. Because the Pearson's correlation would not work. Please specify the nature of your variables before I proceed with my contributions.

Clearly together, most likely in a logistic regression model (if your outcome is binary). Then, you may like to include them or not depending on some metrics of fitness (log likelihood or Bayesian information criteria that accounts for model complexity).

Hi Ali, I have worked a lot with these scales, particularly in either the form of the 25-item Hopkins Symptom Checklist or the longer version the SCL-90, but also other forms. You are right that they are generally highly correlated. Hefin has given you an excellent answer! All I would add is that the approach also depends on your purpose. If all you want to do is account for influence of the variables, so you can examine the influence of somethng else, it is fine to leave them in as separate variables. Also, it is the case, particularly with the HSCL and the SCL-90 that the items are summed into one scale and termed something like "psychological distress." Perhaps you are not interested in the separate results of anxiety and depression and would be satisfied with the role of psychological distress. In the case of anxiety and depression, especially if you use either the HSCL or SCL-90, you would be on solid ground using such a composite. Bob

If two variables are statistically related, then they can not together appear in a linear regression model as independent variables.

Please see also

http://psycnet.apa.org/journals/abn/110/1/40/

for a discussion of that problem

Although you have not explicitly provided a hypothesis you are trying to test, my first thought is: Assuming anxiety and depression are measured as a continuous variables, you need to include BOTH variables in the model. The observed correlation indicate they are collinear (multicollinearity) so modeling one without the other will lead to a distorted understanding (statistical & theoretical) of both when modeled together. Additionally, there are other considerations you need to consider when a linear regression model has moderate or high multicollinearity.

If anxiety and depression are measure as classification variables, you should still include them both as predictors, but there may be several relevant dummy or indicator coding decision you need to consider/make.

In both, how you consider anxiety and depression fundamentally rests on the hypothesis(es) you are testing.

Yes. I these are considered as independent factors they must entered into the regression equation simultaneously. Then the regression coefficients can be compared to know which one has a stronger correlation with the outcome variable.

To avoid augmenting scores because of multicollinearity, you should create two separate models: one with depression and another with anxiety score, and then, you should compare betha estimates of both models and chose the better one. Good luck!

Creating a model for anxiety , one for depression , and one for mixed anxiety and depression with a control to allow a more rigorous study as mixed features are very different from depressive state or anxiety disorder. There is valuable info. Would be great if you could monitor immunological variables.

Great work, best

Ss

There are two school of thoughts: One is you develop two independent models for each variable anxiety and depression and compare both based on goodness of fit; which model explains most of the variability will be preferred. This approach is only acceptable once both variables are highly correlated with rho being greater than 90% (multicollinearity situation).

Second, to develop a mixed model, both anxiety and depression as predictors in a multivariate regression model to adjust for each others effect is way to go in case collinearity is not an issue with rho

Dear all colleagues

With many thanks for your responses, for more information:

- anxiety and depression were assessed using the HADS, a 14-item questionnaire (7 for anxiety and 7 for depression) with 0-3 Likert responses. Score for each anxiety and depression scale ranges from 0 to 21, and the cut-off is 8.

- sleep quality was assessed with PSQI, a 18-item questionnaire which evaluates seven components of sleep quality. The global score ranges from 0 to 21, where scores of > 5 indicates poor sleep quality.

based on your suggestions:

- in the real world, anxiety and depression are not distinct and highly correlated especially in patients with chronic medical conditions

- anxiety and depression may have different effects on various types of sleep disorders

- because we consider the global sleep quality score as the dependent variable, it is better to use a composite score of anxiety and depression as the independent variable

into the model

- in addition, we conducted analyses including both anxiety and depression as independent variables and each time one component of sleep quality as dependent variable. The results were more acceptable according to current evidence.

I would like to answer this question as a psychiatrist and with a clinical approach. As Hefin emphasized, we should look at to what is happen in reality. The so called "anxiety" and "depression" are such phenomena that we do not see them in real clinical situation; what we see is a constellation of symptoms together called, by for example DSM, as "Major Depressive Disorder" or "Generalized Anxiety Disorder". Therefore these variables as dependent ones, are confusing enough to be chosen in a regression analysis.This confusion would become even more complicated when both of them take into account together.

I think for a clinical study, those variables should be chosen that have a greater biological quality and be so independent that could be seen by examiner or stated by patients independent of such vague statements as "I'm depressed". Some suggestions for the domain of "depression" are psychomotor retardation, anhedonia, and diurnal variations or melancholic features, and for "anxiety" are panic attacks, phobias, and psychomotor tension. Choosing genuine parameters is a challenging topic per se.

Dear Aram,

Thanks for your response. We evaluated anxiety and depression with HADS, which is a brief self-administered scale and easily can be used in an outpatient non-psychiatric setting. With acceptable accuracy, HADS can indicate clinically important anxiety and depression. So, we can include them in the regression model as binary variables. Even assuming that we diagnose anxiety and depression with standardized structured psychiatric interview, these two disorders are highly associated and the question is to include them in the regression as separate variables or to include one variable indicative of mood disorder in the model.

Tnx

Anxiety and depression are very close to each other both conceptually and phenomenologically. Anxiety and depression interact in an extremely intricate way, this is not a question of intercorrelation only. What you today would classify as depression you might diagnose as anxiety tomorrow in the same patient. One might ask, how do you include the both sides of a coin in a model simultaneously?

Perhaps consider random forest?

If both measures, anxiety and depression, are entered into the multiple regression, the test of the unique contribution of each (the p value associated with each predictor) will tell you whether when one is present the other adds anything significant to the prediction. If it is their common part that influences sleep, then the unique contribution of both variables will likely be not significant, although either alone would be. This will suggest to you that a composite of the two measures might be still a better predictor than either variable alone. But it could be that what distinguishes depression from anxiety is important to predict sleep quality. Then you would likely have weights of opposite signs for these two variables and the unique contribution of one variable could be more significant in the presence of the other than when alone in the equation.

Since you seem to have multiple sleep quality indicators (you mentioned: "each time one component of sleep quality as dependent variable"), you could consider canonical correlation of your predictor variables on one hand and your sleep components on the other.

You can look at the linear regressions in our paper:

Depression, anxiety, hostility and hysterectomy.

Ewalds-Kvist SB, Hirvonen T, Kvist M, Lertola K,

Psychosom Obstet Gynaecol. 2005 Sep;26(3):193-204.

I would first check out the level of multicollinearity in a linear regression model (vatiance inflation factors). The impact of multicollinearity on prediction variance can be different from its impact on the coefficients. If the overall assessment leads you to conclude that you need to address multicolliearity, one approach is to use Ridge Regression. It will induce a small amount of bias to curb the effects of small eigenvalues due to multicollinearity.